Find the orthogonal projection of b onto a

Question

Given $${\mathbf a} = (-10, -8, 9) $$and $${\mathbf b} = (4, 5, 8) $$ Can someone please define what it means to find the orthogonal projection of b onto a?
Also what is the formula for computing the orthogonal projection of b onto a?
Thank you in advance!
EDIT:
Using the formula for b projection a I get the vectors:
$$(80/245, 64/245, -72/245)$$ But that's incorrect for the orthogonal projection.

Answer 1

The orthogonal projection of a vector $b$ onto a vector $a$ is its component in the direction of $a$. The formula for this is:

$$\mathrm{proj \,\textbf{b}_{\textbf{a}}} = \frac{a \cdot b}{a \cdot a}a$$

This should intuitively make sense. Consider the definition of the dot product in geometric terms, and notice that the projection must be in the direction of $a$.

Now plug your vectors into this formula and get an answer.